Published August 1, 2016
| Version Submitted
Journal Article
Open
A bound for the perimeter of inner parallel bodies
Abstract
We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body Ω. The bound depends only on the perimeter and inradius r of the original body and states that |∂Ω_t| ≥ (1−t/r)^(n−1)₊|∂Ω|. In particular the bound is independent of any regularity properties of ∂Ω. As a by-product of the proof we establish precise conditions for equality. The proof, which is straightforward, is based on the construction of an extremal set for a certain optimization problem and the use of basic properties of mixed volumes.
Additional Information
© 2016 Elsevier Inc. Received 9 November 2015, Accepted 23 February 2016, Available online 3 March 2016. A great deal of gratitude is owed Ari Laptev for many valuable discussions and for proposing the study of the problem at hand. The author would also like to thank Bo Berndtsson for introducing him to convex geometry, and Anders Lundman, Aron Wennman and Eric Larsson for fruitful discussions. Finally we thank the referee for helpful suggestions and comments. The author is supported by Swedish Research Council grant no. 2012-3864.Attached Files
Submitted - 1508.06414.pdf
Files
1508.06414.pdf
Additional details
Identifiers
- Eprint ID
- 102675
- Resolver ID
- CaltechAUTHORS:20200420-153725402
Related works
- Describes
- 10.1016/j.jfa.2020.108574 (DOI)
- https://arxiv.org/abs/1508.06414 (URL)
Funding
- Swedish Research Council
- 2012-3864
Dates
- Created
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2020-04-20Created from EPrint's datestamp field
- Updated
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2023-06-01Created from EPrint's last_modified field